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What Can Oracles Teach Us About the Ultimate Fate of Life?

Salo Ville; Törmä Ilkka

What Can Oracles Teach Us About the Ultimate Fate of Life?

Salo Ville
Törmä Ilkka
Katso/Avaa
LIPIcs-ICALP-2022-131.pdf (1.122Mb)
Lataukset: 

doi:10.4230/LIPIcs.ICALP.2022.131
URI
https://drops.dagstuhl.de/opus/volltexte/2022/16472/
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Julkaisun pysyvä osoite on:
https://urn.fi/URN:NBN:fi-fe2022091258485
Tiivistelmä

We settle two long-standing open problems about Conway’s Life, a two-dimensional cellular automaton. We solve the Generalized grandfather problem: for all n ≥ 0, there exists a configuration that has an nth predecessor but not an (n+1)st one. We also solve (one interpretation of) the Unique father problem: there exists a finite stable configuration that contains a finite subpattern that has no predecessor patterns except itself. In particular this gives the first example of an unsynthesizable still life. The new key concept is that of a spatiotemporally periodic configuration (agar) that has a unique chain of preimages; we show that this property is semidecidable, and find examples of such agars using a SAT solver.

Our results about the topological dynamics of Game of Life are as follows: it never reaches its limit set; its dynamics on its limit set is chain-wandering, in particular it is not topologically transitive and does not have dense periodic points; and the spatial dynamics of its limit set is non-sofic, and does not admit a sublinear gluing radius in the cardinal directions (in particular it is not block-gluing). Our computability results are that Game of Life’s reachability problem, as well as the language of its limit set, are PSPACE-hard.

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